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Vol.
125
(2014)
ACTA PHYSICA POLONICA A
No. 4-A
Acoustic and Biomedical Engineering 2014
Frequency Discrimination for Amplitude Modulated
Sinusoidal Signals at High Carrier Frequencies
A. S¦k
a,∗
and M. Kordus
b
Institute of Acoustics, Faculty of Physics, Adam Mickiewicz University
Umultowska 85, 61-614 Pozna«, Poland
b
Department of Biophysics, Pozna« University of Medical Science, Fredry 10, 61-701 Pozna«, Poland
a
The current study is a continuation of experiments presented by S¦k and Bukaªa ( Acta Physica Polonica A 123,
1106 (2013). The purpose of the present study was to investigate frequency discrimination of amplitude modulated
high frequency carriers. Using 2AFC procedure, the subjects were presented with two observation intervals of which
the rst interval contained four pulses of the same high frequency signal (called SSSS), while in the second interval
(called SHSH) the second and fourth pulses had higher frequencies values (i.e. shifted upwards by ∆f ). The carrier
frequency (in S pulses) was xed and equal to 10 kHz. Modulation rates were equal to 100, 200, 337, 500, 600, 733,
and 800 Hz. The value of the modulation rate was limited to keep all components of the sinusoidal modulation
within one auditory lter (∼ 17% of the center frequency) centered at the carrier frequency. Two dierent types of
modulation were used: a simple sinusoidal modulation with the modulation depth m set to 100%, and a logarithmic
modulation with the modulation depth m set to 50 dB. Results indicate a strong relationship between frequency
discrimination threshold and modulation type. The thresholds are signicantly higher for logarithmic modulation
in comparison to sinusoidal modulation. Amplitude modulation as well as logarithmic modulation applied to the
high frequency carrier cause signicant increase in the frequency discrimination threshold. For high frequency
sinusoidal signal carriers (i.e. close to 10 kHz), frequency discrimination thresholds do not depend on amplitude
modulation rates up to about 800 Hz. In general, the excitation pattern mechanism was a primary cue enabling
frequency discrimination of modulated and unmodulated signals to compare with the mechanism based on the
temporal ne structure. However, the excitation pattern was not the only mechanism responsible for the frequency
discrimination.
DOI: 10.12693/APhysPolA.125.A-149
PACS: 43.66.Fe, 43.66.Hg
1. Introduction
It has been assumed that the frequency discrimination
by normal-hearing subjects is mainly based on temporal
information cues derived from phase locking for frequencies up to 45 kHz [16] whereas above 45 kHz, discrimination is thought to depend on place mechanisms,
based on changes in the excitation pattern [2, 5]. The
supporting arguments come from the behavioral studies [7]. However, some physiological studies by Heinz
et al. [8, 9], Recio-Spinoso et al. [10] and Temchin et
al. [11] suggested the existence of `residual assignment'
of neural impulses to always the same phase of the signal (for frequencies above 5 kHz). Earlier experiments
related to the use of phase locking information by More
and S¦k [12] have shown that the majority of subjects
was able to discriminate between H and I sounds for
F0 = 8001000 Hz with the center frequency of the bandpass lter of 11.1 kHz and 14 kHz (for details, see [12]).
It has been shown that the H and I sound discrimination
could not be based neither on the changes in the signal
spectrum nor on changes in the excitation pattern. The
results indicated that the information about the signal
frequency comes from the time distribution of the peaks
∗ corresponding author; e-mail:
[email protected]
occurring within maxima of amplitude envelope and time
distribution of neural pulses. This suggests that the neural impulses synchronization with specic phase of an
acoustic wave may reach frequencies about 10 kHz or
higher than the previously assumed 5 kHz [8, 12]. However, this statement is controversial taking into account
the well established knowledge on the physiology of hearing. Earlier studies on discrimination of sinusoidal signals by Ritsma et al. [1315] have shown that the detection threshold deteriorates with an increase in frequency
above 5 kHz [14]. However, it is dicult to transfer the
conclusion drawn from those experiments onto complex
signals. On the other hand, the use of phase locking information for frequencies above 5 kHz is recently an important issue [16] considered also in the context of speech
intelligibility for dierent languages [1719] or perception
of musical intervals. For example, Oxenham et al. [16]
have shown that the components of complex tones above
6 kHz produce clearly audible musical pitch and intervals
that must involve the phase locking.
In the present study, the frequency discrimination
experiment was carried out for high frequency carrier
(10 kHz) amplitude modulated at dierent rates. To be
able to see an eect of amplitude envelope superimposed
on the carrier, the frequency discrimination thresholds
were initially determined for high frequency sinusoidal
signal. Then, the same thresholds were determined for
(A-149)
A-150
A. S¦k, M. Kordus
high frequency carrier being amplitude modulated at different rates. As the phase locking seems to provide more
usable information for instantaneous values situated at
extreme values of amplitude envelope, two dierent types
of modulation were used, i.e. sinusoidal and logarithmic
(see below for exact equations). The most important difference between these types of modulation is that the
logarithmic one produces more `peaky' envelope with a
higher crest factor having at the same time more spectral sidebands. It was expected that the amplitude modulation might lead to an improvement of the frequency
discrimination, especially for the amplitude modulator
characterized by a higher crest factor.
2. Aim of the study
The frequency discrimination of the H and I sounds
presented by Moore and S¦k [12] and S¦k and Bukaªa [20]
suggested an eective use of a residual phase locking connected with instantaneous extreme values. On the other
hand, the results suggested an eective use of the temporal ne structure included in the signals, especially when
extreme values coincided with maxima in amplitude envelope. Therefore, the question whether introducing a
sinusoidal envelope on a high frequency carrier may inuence the frequency discrimination of the carrier seems
to be quite crucial. To answer this question was the main
purpose of the presented paper.
3. Materials and method
3.1. Method
The study was based on a modied computer program
and method presented by S¦k and Moore [21]. The modication allowed to generate the amplitude modulated
high frequency carrier using dierent modulators. In accordance with the 2AFC procedure, the subjects were
presented with two observation intervals in a random order. One interval consisted of four identical high frequency pulses (marked as SSSS) while the other observation interval (marked as SHSH), consisted of four pulses
of dierent frequency. However, the second and forth
pulse in that interval had higher frequencies (i.e. shifted
upwards by ∆f ). When amplitude modulation was used,
the identical modulation (i.e. depth and rate) were used
with respect to all pulses in the rst and in the second interval. Each pulse in the SSSS and SHSH tones lasted for
200 ms including 20 ms of rise and fall time (cosine square
function) and was separated from the adjacent sound (S
or H) with a 100 ms interval of silence. The time between
observation intervals was 300 ms. The overall level of signals was equal to 50 dB SL and was adjusted based on
absolute threshold measured prior to the experimental
session. The carrier frequency (in S pulses) was xed
and equal to 10 kHz. Two dierent types of modulation
were used. The rst one was a simple sinusoidal modulation with the modulation depth m set to 100% that can
be described by means of the following equation:
a(t) = (1 + m sin(2πfm t))(sin 2πfc t),
(1)
where fm and fc are the modulation rate and the carrier frequency, respectively. The logarithmic modulation
mentioned above can be described by the following formula:
r sin(2πfm t+θ)
40
sin(2πfc t),
(2)
a(t) = 10
where r denotes modulation depth that was set to 50 dB.
Examples of the above signals are depicted in Fig. 1.
The top panel presents a portion of a waveform obtained
by means of sinusoidal amplitude modulation of a 10-kHz
carrier at a rate of 500 Hz, while the lower one shows logarithmic modulation for modulation depth set to 50 dB.
As can be seen from this graph, the logarithmic modulation results in more `peaky' waveform and contains
smaller number of extreme instantaneous values within
maxima of amplitude envelope. Therefore one may expect that this waveform may produce very prominent
time intervals between neural spikes correlated with extreme instantaneous values of the signal. Logarithmic
modulation results also in much higher crest factor (CF)
which is also specied in the respective panels.
Modulation rates used in the experiment were 100, 200,
337, 500, 600, 733, and 800 Hz. The value of the modulation rate was limited to keep all components of the
sinusoidal modulation within one auditory lter (∼ 17%
of the center frequency) centered at the carrier frequency.
This limitation enabled a direct comparison of the results
acquired for unmodulated and (sinusoidally) modulated
signals, as the spectra of both types of signal were falling
within one auditory lter. The spectrum of the logarithmic modulation that contains more than two sidebands
(especially for chosen modulation depth) and the results
acquired using this type of modulation will be considered
separately.
Fig. 1. The top panel presents example of a waveform
of a 10-kHz carrier with sinusoidal amplitude modulation at a rate of 500 Hz, while the lower one shows logarithmic modulation for modulation depth set to 50 dB
(see detailed equations in text).
Frequency Discrimination for Amplitude Modulated Sinusoidal Signals at High Carrier Frequencies. . .
In the experiment, the task of the subject was to identify the interval which included changes concerning the
presentation of the SHSH series. The experiment began
with a high, clearly audible value of ∆f , so as to allow
the subjects to discriminate the stimuli presented in the
initial stage of each experimental run. Two successive
correct responses brought about a reduction of the frequency dierence, ∆f , and one incorrect answer brought
about an increase of the dierence. Eight turnpoints were
determined in a single experimental run. The change in
the frequency dierence, ∆f , was multiplied (or divided)
by 1.253 until the rst turnpoint, then by 1.252 up to
the second turnpoint, and by 1.251 for the remaining six
turnpoints on which the threshold was calculated. The
threshold values were obtained on the basis of ve independent measurements.
3.2. Subjects
Seven subjects (age up to 27) took part in this experiment. All of them were paid for their services. They
had normal hearing in the range up to 14 kHz which was
analyzed by using Grason-Stadler GSI61 audiometer. All
subjects were given a series of training sessions for about
46 hours. This was done to familiarize the subjects with
their task and to make the test procedure more clear to
them.
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type, three patterns of result accompanied by dierences
in subjects' performance were observed. In general, the
data acquired for subjects S1 and S2 are characterized
by a low thresholds ∆f ranging from 20 Hz to 40 Hz,
while for subjects S3 and S4 the thresholds are within
the range 80100 Hz. The highest discrimination thresholds, ranging from 90 Hz to 130 Hz, were observed for
subjects S5, S6, and S7. Based on the presented raw
data it is very dicult to draw a consistent conclusion
about dependence of the discrimination threshold on the
modulation rate. The results averaged over all subjects
suggest, however, that the modulation rate did not inuence markedly the threshold. Moreover, it seems clear
that the dierence between modulation types may be signicant. It is worth to add that, on average, the obtained
results are broadly consistent with the data collected and
published by Wier et al. [22], as well with the data by
presented by S¦k and Moore [5].
3.3. Apparatus
The signals were generated using the Sound Blaster
Audigy audio card with a 24 bit resolution at 48 kHz
sampling rate controlled by a PC with installed software
proposed by S¦k and Moore [21]. The signals were presented monaurally via ER-2A insert earphones to the ear
chosen by the subject. During the study, the subjects
were in a soundproof booth with the computer peripheries necessary for answering questions concerned with
testing.
4. Results
Results of the frequency discrimination test are presented in Fig. 2. Thresholds are plotted as functions of
amplitude modulation rate. Each panel presents the data
for one subject. The lower right panel depicts thresholds averaged over all subjects. Open squares in each
panel represent the frequency discrimination thresholds,
∆f , for signals being amplitude modulated by means of
a sinusoidal signal (linear modulation), while open circles
represent the thresholds for logarithmic modulation. Vertical lines denote ± one standard error calculated based
on ve repetitions of measurement for each set of parameters (i.e. modulation rate and modulation type). Some
of the datapoints were shifted in frequency domain to
avoid overlapping. The frequency discrimination thresholds acquired for high frequency unmodulated sinusoid
are presented by means of shaded horizontal bars in each
panel. Heights of the bars represent areas of ± one standard error of actual values of the thresholds whose exact
values are situated (but not plotted) in the middle (along)
each bar.
The frequency discrimination threshold for all types of
signals is characterized by a variability across the subjects. For the linear and the logarithmic modulation
Fig. 2. The frequency discrimination thresholds, ∆f ,
for unmodulated and amplitude modulated sinusoidal
signals at high frequencies as functions of the modulation rate. Open squares represent ∆f for linear modulation, open circles for logarithmic one. Vertical lines
denote ± one standard error. The frequency discrimination thresholds for unmodulated signal are presented
by means of shaded horizontal bars. The height of the
bars represents areas of ± one standard error of actual
value of the thresholds.
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A. S¦k, M. Kordus
Raw results obtained for modulated signals, i.e. just
noticeable frequency dierence, were subjected to an
analysis of variance (ANOVA) using GENSTAT statistical package [23]. A within-subject two-way analysis
of variance was carried out with the factors of modulation type (including NO_ AM case) and modulation
rate. The within-subject AVOVA treats the data collected for each subject as a repetition of a single measurement and allows to draw more general conclusion about
the analyzed phenomenon. It turned out that modulation
rate was not statistically signicant [F (5, 30) = 0.953,
p = 0.462]. However, the modulation type was statistically signicant [F (2, 12) = 8.666, p = 0.0005]. The
interaction of the modulation rate and the modulation
type was not statistically signicant [F (10, 60) = 0.96,
p = 0.487] suggesting that the dierences between the
discrimination thresholds for modulated sounds did not
change with the modulation rate.
The post-hoc Tuckey test revealed that the data obtained for all modulation rates created a single homogenous group and there were no statistically signicant differences between averaged data for any two modulation
rates. This proves that, on average, the discrimination
thresholds did not depend on the modulation rate. However, the same test applied to dierent modulation types
(including NO_AM case) revealed statistically signicant dierences between each of two (out of three) groups
of the data, i.e. (AM_100% vs. AM_ 50_dB, p = 0.003;
AM_100% vs. NO_AM, p < 0.001; AM_50_dB vs.
NO_AM, p < 0.001). The lowest frequency discrimination thresholds were obtained for unmodulated sounds
while the highest ones for the logarithmic modulation.
Based on the above presented analysis it can be stated
that the frequency discrimination thresholds for high frequency carrier being amplitude modulated do not depend
on the modulation rate but they do depend on the modulation type, including the NO_AM case. This sort of
result is somehow surprising as initially it was suggested
that the temporal ne structure combined with the phase
locking would provide some additional cues in frequency
discrimination. This should make the subjects' performance slightly better, especially in case of extremes of
instantaneous values coinciding with maxima in amplitude envelope, what actually happened while presenting
modulated signals. However, such improvement was not
observed: the frequency discrimination thresholds were
higher when modulation was introduced on a high frequency carrier.
Therefore, searching for an explanation of this experimental nding, the excitation pattern model [24] was
explored and used. If the dierences in the excitation
patterns evoked by H and S (calculated separately for
each type of modulation) signals would correspond with
the measured averaged thresholds, then the excitation
pattern mechanisms were the primary cue in frequency
discrimination of high frequency sounds.
Using the model described by Moore et al. [25], the
dierences in excitation patterns evoked by the S and H
signals (for the averaged discrimination thresholds ∆f ),
were calculated. Standard MATLAB function (psd [26])
was used to analyze spectral structure of signals with
a resolution of 2 Hz. The spectra were treated as an
input for the calculation of excitation patterns [25]. The
statistical analysis of the acquired data showed that the
discrimination thresholds were, on average, independent
of modulation rate being at the same time signicantly
higher for modulated sounds.
Detection/discrimination cue based on the excitation
pattern dierences was initially suggested by Zwicker and
Fastl [27]. They argued that if the dierence of excitation patterns caused by two signals in any frequency area
is greater than 1 dB, than the signals should be assessed
as dierent. Therefore, the calculation of excitation patterns evoked by S and H signals was done, taking into
account the averaged discrimination threshold, ∆f , both
for the unmodulated signal and separately for each type
of modulated sounds.
As the results of dierences in excitation patterns
across all modulation rates were very similar, an example
result, for modulation rate of 733 Hz, in the frequency
range above 7 kHz, is shown in Fig. 3.
The top line of the gure shows excitation pattern for
S (solid line) and H (dashed line) signals, respectively,
while the bottom row presents the dierences between
these excitations. Subsequent columns of the gure illustrate the data for (unmodulated) sinusoidal signal and
two dierent types of modulation. As it can be seen from
the bottom panels of Fig. 3, for all signal types the difference in excitation patterns reaches values higher than
1 dB.
Fig. 3. Excitation patterns for S (solid line) and H
(dashed line) signals (upper row) and dierences in respective excitation patterns (lower row). Subsequent
columns illustrate data for (unmodulated) sinusoidal
signal and two dierent modulation types (i.e. linear
and logarithmic modulation, respectively).
The model described by Moore et al. [25] took into
account properties of the basilar membrane (i.e. its
tonotopic organization) and provided excitation patterns
Frequency Discrimination for Amplitude Modulated Sinusoidal Signals at High Carrier Frequencies. . .
(in dB) on a logarithmic frequency scale at its output.
This way it is possible to calculate overall excitation pattern dierences (in certain frequency band) in some arbitrary units.
The overall excitation pattern dierences, i.e. the differences summed up in all analyzed frequency bands, are
listed as numbers (in arbitrary units) in each panel of
the bottom row. They reach the highest values for logarithmic modulation being at the same time the lowest
for unmodulated sounds. The order of the overall excitation pattern dierences reects the obtained averaged
results. This suggests that the excitation pattern dierences could be a primary cue in the frequency discrimination. However, if the excitation pattern dierences
were the only detection cue in the frequency discrimination task, the summed up dierences should reach much
closer values. Marked dierences between them suggest
that apart from the excitation pattern cues, some other
mechanism(s) must have been involved in the frequency
discrimination task.
5. Discussion
Results presented by Moore and S¦k [12] and S¦k and
Bukaªa [20] suggested that subjects could use residual
phase locking connected with instantaneous extreme values of sound pressure situated in the maxima of amplitude envelope. This conclusion was drawn based on
the analysis of the output of a single auditory lter centered at a high frequency. The output signal had a very
prominent amplitude envelope that was similar to amplitude modulation. Moreover, the above-quoted authors
stated that the excitation pattern dierences produced
by H and I sounds did not play any role in the discrimination. Therefore, the question whether introducing a
sinusoidal envelope on a high frequency carrier may inuence the frequency discrimination threshold of the carrier
seemed to be quite crucial and was the main purpose of
the present paper. It is well known that a single auditory neuron is capable to generate no more than several
hundred pulses per second [28]. Each pulse is caused
by a temporal depolarization of the inner hair cell [29].
The probability of occurrence of an impulse is monotonically dependent on the amplitude of the stimulating
waveform [30]. Therefore, generation of an impulse seems
to be more probable for extreme instantaneous values in
the maximum amplitude envelope. Impulses do not appear in the adjacent extreme values of the envelope when
its repetition rate is higher than a few hundred times
per second. However, each impulse coincides with a local maximum of an envelope. Innervation of the basilar
membrane is very dense, and then there are at least several aerent neurons that innervate each of the inner hair
cells. One may expect therefore that each maximum of
amplitude envelope will have its unique representation
in the series of pulses observed in a group of neurons.
Time distances between adjacent spikes being close to
successive maxima of envelope but assigned directly to
the extreme values of instantaneous values provide precise information about frequency of the signal. In case
A-153
on an unmodulated sinusoidal signal, situation is quite
dierent. The amplitude envelope is constant and therefore neural spikes may be expected at any time when an
instantaneous value maximum occurs. There will be no
prominent (i.e. the most frequently repeated) time interval that could be used to assess the frequency of the
signal. The distribution of the intervals will be approximately uniform within certain range. Therefore the frequency discrimination of a high frequency unmodulated
sinusoid should be worse.
Results of the present experiment have shown that imposing amplitude modulation on a high frequency sinusoidal waveform did not improve frequency discrimination. Rather an opposite eect was observed: the lowest thresholds were observed for unmodulated sinusoidal
signal. Moreover, the thresholds for logarithmic modulation, the one that produces very prominent peaky amplitude envelope, reached the highest values. This means
that temporal ne structure and the neural pulses associated with the maxima in instantaneous values, located in the extreme values of the amplitude envelope,
even if they would provide additional information related to the frequency of the signal, did not make the
frequency discrimination task easier. Similar situation
was also observed for sinusoidal amplitude modulation,
i.e. the modulation that produces less peaky envelope,
with more extreme instantaneous values within maxima
in the amplitude envelope. The frequency discrimination
thresholds for this type of amplitude modulation were
signicantly lower than those obtained for logarithmic
modulation but on the other hand, signicantly higher
than those observed for unmodulated sounds. In general,
the phase locking was not a primary cue that enabled the
frequency discrimination in the experiment presented in
this paper.
The other possible mechanism that could provide effective information enabling the frequency discrimination
is the excitation pattern mechanism. The analysis of differences in the excitation patterns for signals S and H,
for frequency shift corresponding to average discrimination threshold showed similar results. For all modulation
rates the result was similar, i.e. the highest overall difference in excitation patterns summed up in frequency
bands above 7 kHz were obtained for the logarithmic
modulation, while the lowest one for the unmodulated
sounds. Therefore one can say that a certain correlation between the excitation pattern dierences and the
frequency discrimination thresholds was observed. However, it is dicult to state that the excitation pattern
was the only or a primary mechanism that enabled the
frequency discrimination. If the excitation pattern mechanism were the only mechanism responsible for the discrimination, then the dierences in excitation pattern at
threshold should be very similar but actually, they were
not. One possible reason for which the excitation pattern mechanism did not provide sucient explanation of
the observed eect is the lack of knowledge about frequency resolution of the auditory system in the high fre-
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A. S¦k, M. Kordus
quency area, especially about the auditory lter shape
(symmetry) and bandwidth. Direct measurements of the
psychophysical tuning curves (PTCs) [31, 32] characterizing the auditory lter shape by means of the notched
noise method [33] would provide more detailed information concerning this issue. In summary, the presented
results indicate that the frequency discrimination of high
frequency sounds is not mediated by a single mechanism,
such as the excitation pattern or phase locking, alone.
6. Conclusions
The results of the experiment presented in this paper
allow to formulae the following conclusions:
1. Frequency discrimination thresholds for high frequency (i.e. close to 10 kHz) sinusoidal signal are approximately independent of amplitude modulation rate
up to 800 Hz.
2. Amplitude modulation (both sinusoidal at 100% and
logarithmic at 50 dB) applied to the high frequency carrier brings about a signicant increase in the frequency
discrimination threshold.
3. The frequency discrimination threshold strongly depends on modulation type. The threshold is signicantly
higher for logarithmic modulation compared to sinusoidal
one.
4. Results do not indicate that the temporal ne structure plays a signicant role in frequency discrimination
of amplitude modulated carrier at frequencies close to 10
kHz.
5. The excitation pattern was not the only mechanism
that enabled the frequency discrimination of modulated
and unmodulated signals. However, a certain correlation between the excitation pattern dierences and the
frequency discrimination thresholds for each signal type
was observed.
Acknowledgments
This work was supported by a grant from the from the
National Science Centre No. N N 518 502139.
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